Which term of the AP $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots$ is its first negative term?
The given $\mathrm{AP}$ is $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots .$
Here, $a=20$ and $d=19 \frac{1}{4}-20=\frac{77}{4}-20=\frac{77-80}{4}=-\frac{3}{4}$
Let the nth term of the given AP be the first negative term. Then,
$a_{n}<0$
$\Rightarrow 20+(n-1) \times\left(-\frac{3}{4}\right)<0 \quad\left[a_{n}=a+(n-1) d\right]$
$\Rightarrow 20+\frac{3}{4}-\frac{3}{4} n<0$
$\Rightarrow \frac{83}{4}-\frac{3}{4} n<0$
$\Rightarrow-\frac{3}{4} n<-\frac{83}{4}$
$\Rightarrow n>\frac{83}{3}=27 \frac{2}{3}$
$\therefore n=28$
Hence, the 28th term is the first negative term of the given AP.
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